Bead-spring Chains Research Articles

Anisotropic and enhanced self-diffusion of a macromolecular chain under simple shear flow as revealed by Monte Carlo simulation on lattices

Two-dimensional simple shear flow of a self-avoiding macromolecular chain is simulated by a lattice Monte Carlo (MC) method with a pseudo-potential describing the flow field. The simulated velocity profile satisfies the requirements of simple shear flow unless the shear rate is unreasonably high. Some diffusion problems for a free-draining bead-spring chain with excluded volume interaction are then investigated at low and relatively high shear rates. Three diffusion coefficients are defined and examined in this paper: the conventional self-diffusivity in zero field, Dself, the apparent self-diffusivity in flow field, Dapp, and the flow diffusivity in simulation, Dflow reflecting actually the transport coefficient. It is found that these three diffusivities for a flexible chain are different from each other. What is more important is that self-diffusion exhibits a high anisotropy in the flow field. The apparent self-diffusion along the flow direction is enhanced to a large extent. It is increased monotonically with the increase of shear time or shear strain, whereas the chain configuration can achieve a stationary anisotropic distribution following an interesting overshoot of the coil shape and size. Besides a single self-avoiding chain, an isolated Brownian bead and a group of self-avoiding beads with a quasi-Gaussian spatial distribution are also simulated. According to the comparison, the effects of the connectivity of the chain on the diffusion behavior are revealed. Some scaling relations of Dapp versus t are consistent with the theoretical analyses in the pertinent literature.

Anisotropic and enhanced self-diffusion of a macromolecular chain under simple shear flow as revealed by Monte Carlo simulation on lattices

Two-dimensional simple shear flow of a self-avoiding macromolecular chain is simulated by a lattice Monte Carlo (MC) method with a pseudo-potential describing the flow field. The simulated velocity profile satisfies the requirements of simple shear flow unless the shear rate is unreasonably high. Some diffusion problems for a free-draining bead-spring chain with excluded volume interaction are then investigated at low and relatively high shear rates. Three diffusion coefficients are defined and examined in this paper: the conventional self-diffusivity in zero field, Dself, the apparent self-diffusivity in flow field, Dapp, and the flow diffusivity in simulation, Dflow reflecting actually the transport coefficient. It is found that these three diffusivities for a flexible chain are different from each other. What is more important is that self-diffusion exhibits a high anisotropy in the flow field. The apparent self-diffusion along the flow direction is enhanced to a large extent. It is increased monotonically with the increase of shear time or shear strain, whereas the chain configuration can achieve a stationary anisotropic distribution following an interesting overshoot of the coil shape and size. Besides a single self-avoiding chain, an isolated Brownian bead and a group of self-avoiding beads with a quasi-Gaussian spatial distribution are also simulated. According to the comparison, the effects of the connectivity of the chain on the diffusion behavior are revealed. Some scaling relations of Dapp versus t are consistent with the theoretical analyses in the pertinent literature.

Fracture of flexible polymer chains in dilute solution under transient extensional flow

Applying the technique of Brownian dynamics simulation, we have studied the fracture process of flexible polymer chains when they encounter an extensional flow field of transient character. For this purpose, a mathematical model was made of an experimental device used earlier by other authors to study fracture of polystyrene in dilute solution. The polymer/solvent system studied was a very dilute solution in theta conditions. The polymer molecule was modeled as a FENE bead-spring chain, including a modification to allow for chain fracture. The simulation results showed that the fracture yield depended strongly on flow rate and on molecular weight. We have characterized the molecularweight dependence of the critical flow rate which is necessary for fracture to occur. The distribution of the resulting fragments is interpreted in terms of the conformation of the chains prior to fracture.

Thermal conductivity of dilute solutions of chainlike polymers

The Curtiss–Bird phase-space kinetic theory of polymers is used to derive an expression for the thermal conductivity of a dilute polymer solution, with the polymers represented as arbitrary bead-spring models. Then the general expression is specialized to Rouse bead-spring chains (with Hookean springs). The resulting expression contains several momentum-space averages as well as the configuration-space distribution function for the polymer chains. Use is made of the authors’ previous work on the solution of the Fokker–Planck equation for arbitrary bead-spring models to evaluate the momentum-space averages. Then two special cases are considered: (a) the Hookean dumbbell model, in a fluid with velocity gradients, and (b) the Rouse chain model, with the fluid at rest. For the latter, the authors’ previous study of the properties of tensor Hermite polynomials is helpful for solving the partial differential equation for the configurational distribution function for a polymer molecule in a fluid with a constant imposed temperature gradient. It is shown how the Gaussian distribution function is distorted in a nonisothermal system, but this distortion contributes only about 5% to the final value of the thermal conductivity. The results for the Rouse chain are compared with those previously obtained for several dumbbell models.

On the Shape and Rheology of Linear Micelles in Dilute Solutions

We calculate steady state properties of linear (wormlike) micelles modeled as flexible bead-spring chains from both microscopic and mesoscopic models. The latter model is based on an expression for the free energy of Gaussian chains, modified by a term which takes into account a finite scission energy in order to describe micelles, or breakable polymer chains. In equilibrium, the length distribution then depends on two parameters, namely the micellar concentration and the scission energy. Results of this approach are compared both with previous mesoscopic descriptions and Molecular Dynamics (MD) computer simulation results of the FENE-C model of linear micellar solutions (Phys. Rev. E 53 (1995) 2531). The mesoscopic model is extended to describe flow situations. Implications are discussed and compared with NonEquilibrium MD (NEMD) computer simulation results for the length distribution and flow alignment of linear micelles as well as the corresponding rheological behavior. For the case of steady shear flow both models do predict a decrease of the average micellar size with increasing shear rate.

Added stresses because of the presence of FENE-P bead–spring chains in a random velocity field

FENE-P bead–spring chains unravel in the presence of large enough velocity gradients. In a turbulent flow, this can result in intermittent added stresses and exchanges of energy between the chains and the fluid, whose magnitudes depend on the degree of unravelling and on the orientations of the bead–spring chains. These effects are studied by calculating the average behaviour at different times of an ensemble of chains, contained in a fluid particle that is moving around in a random velocity field obtained from direct numerical simulation of turbulent flow of a Newtonian fluid in a channel. The results are used to evaluate theoretical explanations of drag reduction observed in very dilute solutions of polymers.In regions of the flow in which the energy exchange with the fluid is positive, the possibility arises that turbulence can be produced by mechanisms other than the interaction of Reynolds stresses and the mean velocity gradient field. Of particular interest, from the viewpoint of understanding polymer drag reduction, is the finding that the exchange is negative in velocity fields representative of the wall vortices that are large producers of turbulence. One can, therefore, postulate that polymers cause drag reduction by selectively changing the structures of eddies that produce Reynolds stresses. The intermittent appearance of large added shear stresses is consistent with the experimental finding of a stress deficit, whereby the total local shear stress is greater than the sum of the Reynolds stress and the time-averaged shear stress calculated from the time-averaged velocity gradient and the viscosity of the solvent.

Finitely extensible bead spring chains in time-dependent shear flows

Finitely extensible bead spring chains in time-dependent shear flows

Finitely extensible bead spring chains in time-dependent shear flows

The configurational and rheological properties of bead spring chains in time-dependent shear flows are calculated. The finite extensibility is incorporated through the constraint of constant contour length of the chain. Start-up of shear flow yields a stress overshoot, whereas oscillatory shear flow yields the same frequency dependence of the dynamic moduli as the simple bead spring model. The results show that finite extensibility can lead to non-linear rheological behavior of dilute polymer solutions. The influence of preaveraged hydrodynamic interaction on the obtained results is discussed.

Relaxation in a confined and entangled polymer liquid

We investigate the dynamics of a monodisperse polymer melt that is confined to the vicinity of a plane by a harmonic potential. Polymer molecules are represented by harmonic bead-spring chains with fluctuating bead mobility tensors. Mobility fluctuation rates are determined self-consistently from the chain dynamics. The calculation of the viscoelastic shear modulus is mapped onto the solution of a dynamically disordered random walk, in which a walker hops among lattice sites with fluctuating hopping rates. The relevant random walk problems are solved within the effective medium approximation, yielding the shear modulus and coefficient of shear viscosity as functions of chain length, entanglement molecular weight, and strength of the confining potential. Calculations illustrate the competition between entanglement effects, which retard chain dynamics, and the effect of the external field, which promotes relaxation.

Phase Transitions in Dense Lipid Monolayers Grafted to a Surface: Monte Carlo Investigation of a Coarse-Grained Off-Lattice Model

Semiflexible amphiphilic molecules end-grafted at a flat surface are modeled by a bead-spring chain with stiff bond angle potentials. Constant density Monte Carlo simulations are performed varying temperature, density, and chain length of the molecules, whose effective monomers interact with Lennard-Jones potentials. For not too large densities and low temperatures the monolayer is in a quasi-two-dimensional crystalline state, characterized by uniform tilt of the (stretched) chains. Raising the temperature causes a second-order transition into a (still solid) phase with no tilt. For the first time, finite size scaling concepts are applied to a model of a surfactant monolayer, and it is found that the technique in this case again is useful to locate the transition more precisely. For comparison, also a one-dimensional version of the model is studied, and directions for future extensions of this modeling are discussed.

Some aspects of theory and simulations of polymers in steady flows

AbstractWe present a study of various properties of bead spring chains in steady flows. The Langevin equation of the normal modes of the chain is solved by Fourier transformation. From the resulting power spectrum, the autocorrelation functions of all configuration‐dependent quantities can be calculated. In equilibrium, the influence of the bead masses on the short‐time dynamics is discussed. The influence of different flow fields (shear, elongational and Kramers potential flow) on the mean‐square chain dimension is calculated. A comparison with results obtained from non‐equilibrium molecular dynamics and Monte Carlo calculations is made. Finally, the influence of shear flow on the configurational and rheological properties of cyclic polymers and on the excluded volume behavior of linear chains is examined.

Simulation of Fracture of Flexible Polymer Chains in Transient Elongational Flow

Using the Brownian dynamics technique, we have simulated the fracture process of flexible polymer chains subjected to convergent (sink) flow, where the elongational rate is strongly position dependent. The system studied has been a very dilute solution of polystyrene with molecular weight 2 x 10 6 in cyclohexane at a temperature of 35 °C. The polymer was modeled as a bead-spring chain, adequately parametrized as to reproduce real polymer/solvent conditions. The fracture yield varies with the flow rate and is seen to depend to a high degree on the instrumental setup. The distribution of fragments at the end of the convergent flow region is found to be centered around half the initial molecular weight, and we find that there is no fracture below a certain critical flow rate. These results are in qualitative agreement with the experimental results of Reese and Zimm 1 on DNA and Nguyen and Kausch 2 on polystyrene, a finding that confirms the applicability of the Brownian dynamics simulation technique for studying polymer chain fracture.

Transport properties of polymer melts from nonequilibrium molecular dynamics

We report new results for polymer melts of bead-spring chain molecules obtained by nonequilibrium molecular dynamics simulations of a Couette flow. The beads of these molecules are connected by finitely extendable nonlinear springs. A multiple time scales algorithm has been extended to nonequilibrium situations for integration of the equations of motion. Our model fluids exhibit shear thinning and nonzero normal-stress differences. We have performed simulations both at constant volume and at constant pressure for melts of linear chains of up to 50 beads. We find that, at constant pressure, the density of long chain melts increases significantly with shear rate. We also report results of simulations for branched chain models. We find that the length of the side groups largely enhances the viscosity at low shear rates.

Brownian dynamics simulation of bead–spring chain models for dilute polymer solutions in elongational flow

We report results of Brownian dynamics simulations of the steady state and transient rheological behavior of bead and spring chain models in elongational flow. The simulations include examination of the effects of excluded volume, hydrodynamic interactions, chain length, and maximum bond extension. Hydrodynamic interactions are modeled by the Rotne–Prager–Yamakawa tensor. The steady state results for the rheological properties of finitely extensible nonlinear elastic (FENE) chains in the absence of hydrodynamic interactions and excluded volume interactions are compared with the analytic FENE‐PM theory of Wedgewood and co‐workers. For bead and spring models with short bond lengths, excluded volume and hydrodynamic interactions are shown to affect the steady state rheological properties significantly.

Perturbation expansion of polymers in shear flow

AbstractThe effect of shear flow on the excluded volume properties of bead spring chains is calculated via the perturbation expansion of the mean‐square end‐to‐end distance with respect to the excluded volume parameter z. The coefficient of the series 〈R2〉/〈R2〉0 = 1 + C1 · z − … vanishes for large flow rates but shows a strong dependence on the shear rate in the intermediate regime.

Configurational and rheological properties of cyclic polymers

The configurational and rheological properties of multiple-twisted ring polymers, modelled as long cyclic finitely extensible bead–spring chains with internal distance constraints, have been calculated. This model corresponds to chemically cross-linked Rouse chains. The calculations were carried out for equilibrium and steady shear flow. The mean-square radius of gyration in equilibrium depends on the number of loops. In shear flow, non-Newtonian flow behaviour arises. The shear viscosity and the first normal stress coefficient decrease with increasing shear rate. The second normal stress coefficient is negative and much smaller than the first normal stress coefficient. Since the mean-square radius of gyration, shear viscosity and the first and second normal stress coefficients depend on the number of loops, the latter can be determined by four independent experiments.

On consistency criteria for stress tensors in kinetic theory models

Two different consistency criteria are considered for the stress tensor expressions of two different kinetic theory models for dilute polymer solutions. The criteria are material objectivity and thermodynamic consistency. Two separate approaches for checking thermodynamic consistency are considered, the first proposed by Grmela [Phys. Lett. 111A, 41 (1985)] and the second by Jongschaap [Lecture Notes in Physics (Springer, Berlin, 1991)]. The two models considered are a Hookean dumbbell model with internal viscosity (no linearization or rotation matrix approximations), and an inertial, elastic bead-spring chain. Both sets of criteria may be applied to these models without finding either a closed-form expression for the stress tensor, or solving any of the equations. We find that the Giesekus form for the stress tensor expression for a Hookean dumbbell with internal viscosity is both thermodynamically consistent and objective, whereas the Kramers form is not thermodynamically consistent. The Kramers form for the stress tensor is found to be thermodynamically consistent for an inertial, elastic bead-spring chain, but is not materially objective. We also find that the approaches of Grmela and Jongschaap are the same, although each requires a different ansatz for the relationship between thermodynamic quantities and polymer conformations.

Bead-spring chain model for the dynamics of dilute polymer solutions: Part 2. Comparisons with experimental data

Comparisons are made between experimental rheological data and theoretical predictions obtained from a recently developed algorithm which incorporates three major molecular concepts in a theory for dilute polymer solutions (hydrodynamic interaction, excluded volume and nonlinear springs). These predictions include the radius expansion factor, the apparent chain expansion factor, the molecular weight dependence of the intrinsic viscosity, the frequency dependence of oscillatory flow birefringence, and the shear rate dependence of the intrinsic viscosity. This paper shows that a bead-spring chain model quantitatively predicts these quantities when the relevant molecular concepts are incorporated, suggesting that the rheological properties of dilute polymer solutions can be explained and predicted in terms of these molecular parameters.

Finitely extensible bead–spring chain macromolecules in steady elongational flows

Infinitely dilute polymer solutions undergoing elongational flows are studied by modeling the polymer molecules as bead-spring chains. Monte Carlo integration is used to calculate the properties of chains of finitely extensible nonlinear elastic springs in steady uniaxial elongational flow from the exact solution for the distribution of configurations. The properties calculated include the mean square end-to-end distance of the chains and the elongational viscosity and intrinsic birefringence of the solution. These properties are compared to the predictions of the model under the Peterlin approximation. Whereas the models with and without the approximation show qualitatively similar results, there are substantial quantitative differences.

Browian dynamics simulation of finitely extensible bead-spring chains

The behaviour of infinitely dilute solutions of finitely extensible non-linear elastic (FENE) bead-spring chains in shear flow and uniaxial elongational flow is analysed under both steady-state and transient conditions, using the method of Brownian dynamics. Three FENE chain models are considered: the original model using the Warner spring force law and models using the Peterlin approximation and the recently proposed PM approximation. It is shown that the two approximate models, which are based on a pre-averaging of the spring constant, perform reasonably well in elongational flow but fail in describing shear flow.